Question 14.3: A Pain in Your Ear Estimate the force exerted on your eardru...
A Pain in Your Ear
Estimate the force exerted on your eardrum due to the water when you are swimming at the bottom of a pool that is 5.0 m deep.
Conceptualize As you descend in the water, the pressure increases. You may have noticed this increased pressure in your ears while diving in a swimming pool, a lake, or the ocean. We can find the pressure difference exerted on the eardrum from the depth given in the problem; then, after estimating the ear drum’s surface area, we can determine the net force the water exerts on it.
Categorize This example is a substitution problem.
The air inside the middle ear is normally at atmospheric pressure P_0. Therefore, to find the net force on the eardrum, we must consider the difference between the total pressure P_{\text {bot }} at the bottom of the pool and atmospheric pressure. Let’s estimate the surface area of the eardrum to be approximately 1 cm^2=1 \times 10^{-4} m^2.
Use Equation 14.4 to find this pressure difference:
P=P_0+\rho g h (14.4)
\begin{aligned}P_{\text {bot }}-P_0 & =\rho g h \\& =\left(1.00 \times 10^3 kg/ m^3\right)\left(9.80 m/s^2\right)(5.0 m)=4.9 \times 10^4 Pa\end{aligned}Use Equation 14.1 to find the magnitude of the net force on the ear:
P \equiv \frac{F}{A} (14.1)
F=\left(P_{\text {bot }}-P_0\right) A=\left(4.9 \times 10^4 Pa\right)\left(1 \times 10^{-4} m^2\right) \approx 5 NBecause a force of this magnitude on the eardrum is extremely uncomfortable, swimmers often “pop their ears” while under water, an action that pushes air from the lungs into the middle ear. Using this technique equalizes the pressure on the two sides of the eardrum and relieves the discomfort.